Automorphic Forms and Automorphic Vector Bundles
Weixiao Lu
June 2020
Contents
1 Automorphic Forms and Automorphic Representations 1
2 Automorphic Vector Bundles and Automorphic Representations 3
3 Relations to (g,K) Cohomology 5
References 8
Happy Dragon Boat Festival!
1 Automorphic Forms and Automorphic Representations
Denote A = AQ and Af = AQ,f be the finite adele. Let G be a reductive group over Q, Z is the center of G and g is the complexification of the Lie algebra of G(R), U(g) and Z(g) stands for the universal enveloping algebra and the center of universal enveloping algebra respectively.
Kmax,f ⊂ G(Af ) be a maximal compact subgroup, K∞ ⊂ G(R) be a maximal compact subgroup of Lie group G(R). Denote Kmax = Kmax,f × K∞.
Definition 1.1 (Following [1]). A function ϕ : G(A) → C is called an automorphic form if
• ϕ is smooth. i.e. locally constant on the non-archimedean part and C∞ on the archimedean part.
• ϕ is left G(Q) invariant.
1 • ϕ is Kmax finite. i.e. The subspace generated by Kmax · ϕ is finite dimensional. Where the action is given by (kϕ)(g) = ϕ(gk).
• ϕ is Z(g) finite. i.e. ϕ is annihilated by a finite codimensional ideal of Z(g).
• ϕ is moderate growth.
The space of all automorphic forms is denoted by A(G). Let ω : Z(Q)\Z(A) → C× be a central character, define A(G; ω) as follows
A(G, ω) = {ϕ ∈ A(G)|ϕ(zg) = ω(z)ϕ(g), ∀z ∈ Z(A), x ∈ G(A)}
Remark. We will pretend that G(Q)\G(A)/Z(R) is compact, which is equivalent to say that there is no growth condition. Although, in our main example G = GL2 this will be false, but we pretend it is true(We will lost automorphic forms that are not cusp forms, for example, Eisenstein series)
Example 1.2. For G = GL2,Z = Gm,in this case g = gl(2, C) and Z(g) = C[∆,Z], where 1 2 ∆ = EF + FE + 2 H is the Casimir operater and Z is the center of g. Let Γ = Γ0(N) . For a modular form f ∈ Sk(Γ0(N), ψ), we can define a adelic lift
−k ϕf (g) = f(g∞(i))j(g∞, i) ψ(k0)
Which can be checked to be a automorphic form. Moreever, we have an isomorphism
1 Γ\0(N) S (Γ (N)) =∼ A h∆ − (k2 − 1)i, σ k 0 cusp 4 k
(See [2])
Back to general discussion, We have a natural action of G(A) on A(G) or A(G, ω). i.e. these space is natural a (g,K∞) × G(Af ) module. The action is given by
(X, k, g, ϕ) 7→ (g 7→ Xϕ(xkg))
Recall that we have assumed that all automorphic forms are cuspidal, then we have a spectrum decomposition(called the cuspidal spectrum)
M 0 m(π) A(G, ω) = (π∞ ⊗ (⊗lπl)) π Where m(π) is a finite number, called the automorphic multiplicity. ⊗0 is the restricted tensor 0 product. And for all but finitely many finite prime l,πl is unramified principal series. and vl is the sperical vector for πl.
2 (For a proof, see [1] Chapter 9 or [3] Chapter 3, where A is replaced by Acusp) G(Ql) Here unramified principal series means πl = Ind χl. Where B is Borel subgroup and T is B(Ql) the Levi quotient. And × χl : T (Ql) → T (Ql)/T (Zl) → C G(Zl) G(Ql) is an unramified character. As G( l) = B( l)G( l) (Iwasawa decomposition), so if ϕ ∈ Ind χl . Q Q Z B(Ql) Then
ϕ(g) = ϕ(bk) = χl(b)ϕ(1)
G(Zl) 0 So dim πl = 1, we then choose vl = ϕ(1) ! ! ∗ ∗ ∗ 0 Example 1.3. For G = GL2,B = , and T = . 0 ∗ 0 ∗ × ! Q 0 χ is of the form l → × → , (x, y) 7→ αvl(x)βvl(y) l × Z Z C 0 Ql
Definition 1.4. An automorphic representation is a representation of (g,K∞) × G(Af ) of the 0 form π∞ ⊗ (⊗lπl) which occurs in the cuspidal spectrum decomposition(for some ω).
Remark. In general, an automorphic representation is an irreducible admissible subrepresen- tation of natural action of (g,K∞) × G(Af ) on A(G). And automorphic representation defined above is called cuspidal automorphic representation.
2 Automorphic Vector Bundles and Automorphic Represen- tations
Fix an algebraic representation W of K∞. And fix an open compact subgroup Kf ⊂ G(Af ). We have an locally symmetric space(real manifolds)
ShG(Kf ) = G(Q)\ (G(Af )/Kf ) × (G(R)/K∞) and an vector bundle on ShG(Kf ):
K∞ W = G(Q)\ (G(Af )/Kf ) × G(R) × W
So we have
∞ {C sections of W } = {ϕ : G(Q)\G(A) → W, K∞ equivariant, Kf invariant}
Which is roughly equal to (A(G, ω)Kf ⊗ W )K∞ if we omit the cusp issue and choose ω compatible with W .
3 Example 2.1. For G = GL2
List of irreducible admissible (g,K) representation π∞: ([4], Chapter 7 or [3] 2.3):
• holomorphic discrete series:
π∞ = πk ⊕ πk+2 ⊕ · · · ! cos θ sin θ 2πilθ Where k > 0, πl stands for K = SO(2) acts as 7→ e . And the g action − sin θ cos θ is like an “anti” Verma module with integer weights: k k Ev = ( + l)v, F v = ( − l)v l 2 l 2 l−2
• antiholomorphic discrete series:
π∞ = ··· π−k−4 ⊕ π−k−2 ⊕ π−k
where k > 0 and g acts as a Verma module of integral weight.
• principal series
π∞ = ··· πk−2 ⊕ πk ⊕ πk+2 ⊕ · · ·
g action looks like “Verma module with non-integer weight”
• finite dimensional representation
• limit of discrete series(corresponds to weight 1 form)
Fix the representation
× × −k χ−k : K∞ = R · SO(2) → C , (r, z) 7→ z
From previous lecture, we learned that this corresponds to ωk And we have
∞ . Kf K∞ M Kf ⊕m(π) K∞ C (ShG(Kf ),W −k) = (A(G) ⊗ W−k) = (πf ) ⊗ (π∞ ⊗ W−k) π
K∞=χk The last term is just π∞ It has a subspace
M Kf ⊕m(π) K∞=χk F =0 Hol(ShG(Kf ),W −k) = (πf ) ⊗ (π∞ ) π
4 K∞ So k is the lowest weight, by the list of classification above, when π∞ ⊗ W−k) non zero, π∞ must be discrete series representation. We can use above discussion to explain old/new form theory.
M Γˆ K∞=χk Sk(Γ) = Hol(ShG(Γ)b , χ−k) = (πf ) ⊗ π∞ π
(p) 0 0 Consider Γ = Γ1(N), p 6 |N,K = Γb = K GL2(Zp). And Γ ⊃ Γ = Γ1(N) ∩ Γ0(p),K = (p) K Iwp.Where × ! Iw = Zp Zp p × pZp Zp is the Iwasawa subgroup. We have two types of operators, which fits into the following diagram
(p),K(p) L ⊕m(π) GL2(Zp) Sk(Γ) = π(πf ) ⊗ (πp)
(p) 0 L (p),K ⊕m(π) Iwp Sk(Γ ) = π(πf ) ⊗ (πp)
( M (p),K p) ⊕m(π) Iwp GL2(Zp) New forms: = (πf ) ⊗ (πp) ,sum for those π such that πp = 0 but π Iwp Iwp πp 6= 0. Special for GL2 theory, for these π, dim πp = 1.
Old Forms: If πp is an unramified principal series, we have an isomorphism
GL2( p),⊕2 ∼ Iwp (πp) Z = (πp)
3 Relations to (g,K) Cohomology
Now, we move to the general situation: Recall that we have gotten
∞ M Kf ⊕m(π) K∞ {C sections of W } = (πf ) ⊗ (π∞ ⊗ W ) π Let’s assume that we are in the situation of Shimura varieties, we want cohomology for holo- • morphic sections H (ShG(Kf ),W ). Use Dolbeault resolution
1 2 d Ohol C∞ ∂ Ω ∂ Ω ∂ ··· ∂ Ω 0 ShG(Kf )
5 + 1 ∼ + ∗ 1 ∼ − ∗ Note that TCShG(Kf ) = g/q = p ,so Ω = (p ) and Ω = (p ) Tensoring with W , we get a resolution of W hol, which implies
! • hol • M Kf ⊕m(π) • − H (ShG(Kf ),W ) = H (πf ) ⊗ (π∞ ⊗ Hom(∧ (p ),W )) π And the latter term is exactly the (g,K) cohomology we now introduce.
What is (g,K) cohomology?[5]
(1) Lie algebra cohomology Let g be a Lie algebra and V be a g module. Define
q q C (g; V ) = HomkΛ (g),V )
and d : Cq(g; V ) → Cq+1(g,V ) is given by
X i X i+j df(x0, ··· , xq) = (−1) xif(x0, ··· , xbi, ··· , xq)+ (−1) f([xi, xj], x0, ··· , xbi, ··· , xbj, ··· , xq) i i The cohomology is denoted by H•(g,V ). Which is also the right derived functor of V 7→ V g := {v ∈ V |xv = 0, ∀x ∈ g}.i.e. p p H (g,V ) = ExtUg(k, V ) where k is the base field. (2) Relative Lie algebra cohomology Let k ⊂ g be a Lie subalgebra.Define q q C (g, k,V ) = Homk(Λ (g/k),V ) q X = {f ∈ C (g,V )|f only depends on xi ∈ g/k, f(x1, ··· , [x, xi], ··· , xq) = xf(x1, ··· , xq), for x ∈ k} i It can be verified that C•(g, k,V ) is a sub-cochain complex of C•(g,V ), whose cohomology is denoted by H•(g, k,V ). (3)( g,K) cohomology and (q,K) cohomology. Now we work over C, let g be a Lie algebra, not necessrily reductive. Let G be a real Lie group with complexified Lie algebra g and K be the maximal compact with connected component K0. 6 Let V be a (g,K) module, define q q ∼ q K/K0 C (g,K,V ) := HomK (Λ (g/k),V ) = C (g, k,V ) Whose cohomology groups is denoted by H•(g,K; V ). Proposition 3.1 (See [5]). q q K/K0 q H (g,K; V ) = H (g, k; V ) = Extg,K (C,V ) Theorem 3.2. • M Kf ⊕m(π) • H (ShG(Kf ),W ) = (πf ) ⊗ H (q,K; π∞ ⊗ W ) π A deep theorem Theorem 3.3. When π∞ is a discrete series or limit of discrete series, W is irreducible.Then q H (q,K∞; π∞ ⊗ W ) is non zero at exactly one degree if G(R) is connected. And in this case the dimension of the non-vanishing cohomology group is 1. Example 3.4. For G = GL2,π∞ is the discrete series πk ⊕ πk+2 ⊕ · · · We have 0 dim H (q,K∞; π∞ ⊗ W−k) = 1 and 1 dim H (q,K∞; π∞ ⊗ Wk−2) = 1 Example 3.5. Let F be a totally real field, G = ResF/QPGL2 weight k = (kτ )τ∈Hom(F,R), all are even. n0 = #{τ ∈ Hom(F, R), kτ ≤ 0} By the multiplicity one theorem for PGLn • kτ M Kf O • H (ShG(Kf ), ω ) = (πf ) ⊗ H (qτ ,K∞,τ ; πτ ⊗ χ−kτ ) π τ∈Hom(F,R) • And H (qτ ,K∞,τ ; πτ ⊗χ−kτ ) is concentrated in one degree and dim 1. It concentrates in degree 0 if kτ ≥ 2 and in degree 1 if kτ ≤ 0. • kτ So H (ShG(Kf ), ω ) is concentrated in degree n0. Another Relationship between cohomology of vector bundles and (g,K) cohomology When V is an algebraic C representation of G defined over a number field. We can associated a locally constant sheaf V on ShG(Kf ). V = V ⊗C OShG(Kf ) is the deRham local system. We get 7 H• (Sh (K ),V ) = • Sh (K ), V → V ⊗ Ω1 → · · · Ωd → · · · Betti G f H G f ShG(Kf ) ShG(Kf ) • •,• = (ShG(Kf ), V ⊗ Ω ) H ShG(Kf ) M Kf ⊕m(π) • • + • − K∞ = (πf ) ⊗ H π∞ ⊗ Hom(Λ p ⊗ Λ p ,V ) M Kf ⊕m(π) • = (πf ) ⊗ H (g,K∞; π∞ ⊗ V ) Example 3.6. F totally real,G = ResF/QPGL2,F Langlands observation: mid dim H (g,K∞; π∞ ⊗ V (λ)) = dim(representation of Gb of highest weight µ) or rather this is how Langlands discovered the dual group. References [1] Hahn Getz. an Introduction to automorphic representations. [2] Gelbart. Automorphic Forms on Adele Groups. [3] Daniel Bump. Automorphic Forms and Representations. [4] Hunley Goldfeld. Automorphic Representations and L-Functions for the General Linear Group(Volume 1). [5] Wallach Borel. Continuous cohomology, discrete subgroups, and representations of reductive groups. 8